Abundance of entire solutions to nonlinear elliptic equations by the variational method

L. M. Lerman; P. E. Naryshkin; A. I. Nazarov

Abundance of entire solutions to nonlinear elliptic equations by the variational method

Analysis of PDEs 2018-11-09 v1

Authors: L. M. Lerman , P. E. Naryshkin , A. I. Nazarov

Abstract

We study entire bounded solutions to the equation $\Delta u - u + u^3 = 0$ in $\mathbb R^2$ . Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in a unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. The method is also applicable for more general equations in any dimension.

Keywords

entire solutions nonlinear elliptic equations variational method concentration arguments symmetry bounded solutions

Cite

@article{arxiv.1811.03143,
  title  = {Abundance of entire solutions to nonlinear elliptic equations by the variational method},
  author = {L. M. Lerman and P. E. Naryshkin and A. I. Nazarov},
  journal= {arXiv preprint arXiv:1811.03143},
  year   = {2018}
}

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30 pages, 28 figures

Abundance of entire solutions to nonlinear elliptic equations by the variational method

Abstract

Keywords

Cite

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